Partition
A partition of a number is a way of expressing it as the sum of some number of positive integers. For example, the partitions of 3 are: 3, 2+1, and 1+1+1 (notice how the order of the addends is disregarded).
There is no known, simple formula that gives the number of partitions of a number. There is, however, a rather ugly formula discovered by G. H. Hardy, J. E. Littlewood, and Srinivasa Ramanujan. However, this formula is rather unwieldy: it is not even known for which values of n is the number of partitions of n even, despite the presence of a formula!
A more fruitful way of studying partition numbers is through generating functions. The generating function for the partitions is given by 
. Partitions can also be studied by using the Jacobi theta function, in particular the triple product.
